A solid conducting sphere, having a charge `Q`, is surrounded by an unchanged conducting hollow spherical shell. Let the potential difference between the surface of the solid sphere and that of the outer surface of the hollow shell be `V`. If the shell is now given a change of `-4Q`, the new potential difference between the same tow surface is :

Options

A. `-2V`
B. `2V`
C. `4V`
D. `V`

Answer

To solve the problem, we need to analyze the situation step by step.

### Step 1: Understand the Initial Setup
We have a solid conducting sphere with charge \( Q \) and a surrounding conducting hollow spherical shell that is initially uncharged. The potential difference \( V \) is defined between the surface of the solid sphere and the outer surface of the hollow shell.

### Step 2: Determine the Potential of the Solid Sphere
The potential \( V_A \) at the surface of the solid sphere (radius \( r_A \)) is given by:
\[ V_A = \frac{Q}{r_A} \]
This is because the potential inside a conductor is constant and equal to the potential on its surface.

### Step 3: Determine the Potential of the Hollow Shell
Since the hollow shell is initially uncharged, it will have no contribution to the potential at its outer surface. The potential \( V_B \) at the outer surface of the hollow shell (radius \( r_B \)) is influenced by the charge \( Q \) of the inner sphere:
\[ V_B = \frac{Q}{r_B} \]
However, since the shell is uncharged, we consider the potential due to the inner sphere only.

### Step 4: Calculate the Initial Potential Difference
The potential difference \( V \) between the solid sphere and the hollow shell is:
\[ V = V_A - V_B = \frac{Q}{r_A} - 0 = \frac{Q}{r_A} \]

### Step 5: Analyze the Change in Charge of the Hollow Shell
Now, the hollow shell is given a charge of \( -4Q \). The new charge on the shell does not affect the potential difference between the two surfaces because the potential difference depends only on the charge of the inner sphere.

### Step 6: Determine the New Potential of the Hollow Shell
After the shell is charged with \( -4Q \), the potential \( V_B \) at the outer surface of the shell is still influenced by the charge \( Q \) of the inner sphere. The potential due to the charge \( -4Q \) on the shell does not affect the potential difference calculation between the two surfaces.

### Step 7: Conclusion
Thus, the new potential difference between the surface of the solid sphere and the outer surface of the hollow shell remains:
\[ V = \frac{Q}{r_A} - 0 = \frac{Q}{r_A} \]
This means that the potential difference remains unchanged despite the charge on the hollow shell.

### Final Answer
The new potential difference between the surface of the solid sphere and that of the outer surface of the hollow shell is still \( V \).

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